Question

Evaluate: $\int \frac{\log x}{(1+\log x{)}^2}dx$.

Answer 1

$\Rightarrow$ Sub. $x=e^t$ $\therefore dx=e^{t}dt$

$I=\int \frac{log{e}^t}{(1+log{e}^t)}{e}^{t}dt$

$=\int \frac{t{e}^t}{(1+t{)}^2}dt$

Sub. $\frac{e^t}{1+t}=y$

$\therefore \frac{(1+t)e^t-e^t}{(1+t{)}^2}dt=dy$

$\frac{t{e}^t}{(1+t{)}^2}dt=dy$

$\therefore I=\int dy$

$=y+c$

$=\frac{e^t}{1+t}+c$ ($\because$ we have taken the substitution $x=e^t)$

$I=\frac{x}{1+logx}+c$